Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 2 / 12K / a / Solution

A binary cyclic code is a linear subspace $C\le {\mathbb{F}}_2^n$ invariant under cyclic coordinate shift. Identify a word $(c_0,\dots ,c_{n-1})$ with $c_0+c_{1}X+\cdots +c_{n-1}{X}^{n-1}$ inCyclic shift is multiplication by $X$, so cyclic codes are exactly ideals of this quotient. Since ${\mathbb{F}}_2[X]$ is a principal ideal domain, each is generated by a unique monic divisor $g\mid X^n-1$, giving the required bijection.